# The Spectral Representation of Homogeneous Spin-Weighted Random Fields on the Sphere

## TL;DR

This paper derives a spectral representation for homogeneous spin-weighted spherical random fields, extending cosmological models without mode decomposition, and simplifies calculations using Spin(3) group representations.

## Contribution

It provides a general spectral representation for arbitrary integer spin fields on the sphere, unifying and extending previous cosmological results.

## Key findings

- Different modes are generally uncorrelated.
- The power spectrum definition is generalized.
- Spectral representation characterizes homogeneity.

## Abstract

This is a direct computation of the spectral representation of homogeneous spin-weighted spherical random fields with arbitrary integer spin. It generalises known results from Cosmology for the spin-2 Cosmic Microwave Background polarisation and Cosmic Shear fields, without decomposition into $E$- and $B$-modes. The derivation uses an instructive representation of spin-weighted spherical functions over the Spin(3) group, where the transformation behaviour of spin-weighted fields can be treated more naturally than over the sphere, and where the group nature of Spin(3) greatly simplifies calculations for homogeneous spherical fields. It is shown that i) different modes of spin-weighted spherical random fields are generally uncorrelated, ii) the usual definition of the power spectrum generalises, iii) there is a simple relation to recover the correlation function from the power spectrum, and iv) the spectral representation is a sufficient condition for homogeneity of the fields.